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This is a variant of the classic

My barbarian has CHA as his dump stat, but he is way stronger™[sic]. Therefore, I roll Intimidate with STR by showing how strong I am to wimpy kobolds. Break door with my armpit powaaaa!!!

It used to happen in previous editions, but in I am not so sure.

Can a character, given a specific situation where it might be fitting to think of such substitution of relevant ability scores on a skill roll, use a different ability score / modifier?

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    \$\begingroup\$ 5e doesn't have skill checks. It has ability checks to which skills may apply. \$\endgroup\$ – Derek Stucki Jun 21 '17 at 4:46
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Yes, Subject to DM Approval

There is a section in the PHB entitled "Variant: Skills with Different Abilities". The two specific examples they give are using Constitution on an Athletics check to swim an unusually far distance, and:

Similarly, when your half-orc barbarian uses a display of raw strength to intimidate an enemy, your DM might ask for a Strength (Intimidation) check, even though Intimidation is normally associated with Charisma. -PHB p. 175

Naturally, any Ability switching would be subject to DM approval. The rule is a "Variant", and thus might not be adopted by all DMs. Even if it is adopted by your DM, they are the decider for when it is applicable, and it might not be doable for all characters within a group. A DM might rule that you can't use another Ability unless you're proficient in the check, for example. In fact, the RAW only explicitly mentions checks you are proficient in:

In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the DM might ask for a check using an unusual combination of ability and skill, or you might ask your DM if you can apply a proficiency to a different check. -PHB p. 175

But there is RAW support for switching Abilities.

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